Financial Market Volatility

Ser-Huang Poon

A Practical Guide to Forecasting

Financial Market Volatility

For other titles in the Wiley Finance series

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A Practical Guide to Forecasting

Financial Market Volatility

Ser-Huang Poon

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Library of Congress Cataloging-in-Publication Data

Poon, Ser-Huang.

A practical guide for forecasting ¬nancial market volatility / Ser Huang

Poon.

p. cm. ” (The Wiley ¬nance series)

Includes bibliographical references and index.

ISBN-13 978-0-470-85613-0 (cloth : alk. paper)

ISBN-10 0-470-85613-0 (cloth : alk. paper)

1. Options (Finance)”Mathematical models. 2. Securities”Prices”

Mathematical models. 3. Stock price forecasting”Mathematical models. I. Title.

II. Series.

HG6024.A3P66 2005

332.64 01 5195”dc22 2005005768

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-85613-0 (HB)

ISBN-10 0-470-85613-0 (HB)

Typeset in 11/13pt Times by TechBooks, New Delhi, India

Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall

This book is printed on acid-free paper responsibly manufactured from sustainable forestry

in which at least two trees are planted for each one used for paper production.

I dedicate this book to my mother

Contents

Foreword by Clive Granger xiii

Preface xv

1 Volatility De¬nition and Estimation 1

1.1 What is volatility? 1

1.2 Financial market stylized facts 3

1.3 Volatility estimation 10

1.3.1 Using squared return as a proxy for

daily volatility 11

1.3.2 Using the high“low measure to proxy volatility 12

1.3.3 Realized volatility, quadratic variation

and jumps 14

1.3.4 Scaling and actual volatility 16

1.4 The treatment of large numbers 17

2 Volatility Forecast Evaluation 21

2.1 The form of X t 21

2.2 Error statistics and the form of µt 23

2.3 Comparing forecast errors of different models 24

2.3.1 Diebold and Mariano™s asymptotic test 26

2.3.2 Diebold and Mariano™s sign test 27

2.3.3 Diebold and Mariano™s Wilcoxon sign-rank test 27

2.3.4 Serially correlated loss differentials 28

2.4 Regression-based forecast ef¬ciency and

orthogonality test 28

2.5 Other issues in forecast evaluation 30

viii Contents

3 Historical Volatility Models 31

3.1 Modelling issues 31

3.2 Types of historical volatility models 32

3.2.1 Single-state historical volatility models 32

3.2.2 Regime switching and transition exponential

smoothing 34

3.3 Forecasting performance 35

4 Arch 37

4.1 Engle (1982) 37

4.2 Generalized ARCH 38

4.3 Integrated GARCH 39

4.4 Exponential GARCH 41

4.5 Other forms of nonlinearity 41

4.6 Forecasting performance 43

5 Linear and Nonlinear Long Memory Models 45

5.1 What is long memory in volatility? 45

5.2 Evidence and impact of volatility long memory 46

5.3 Fractionally integrated model 50

5.3.1 FIGARCH 51

5.3.2 FIEGARCH 52

5.3.3 The positive drift in fractional integrated series 52

5.3.4 Forecasting performance 53

5.4 Competing models for volatility long memory 54

5.4.1 Breaks 54

5.4.2 Components model 55

5.4.3 Regime-switching model 57

5.4.4 Forecasting performance 58

6 Stochastic Volatility 59

6.1 The volatility innovation 59

6.2 The MCMC approach 60

6.2.1 The volatility vector H 61

6.2.2 The parameter w 62

6.3 Forecasting performance 63

7 Multivariate Volatility Models 65

7.1 Asymmetric dynamic covariance model 65

Contents ix

7.2 A bivariate example 67

7.3 Applications 68

8 Black“Scholes 71

8.1 The Black“Scholes formula 71

8.1.1 The Black“Scholes assumptions 72

8.1.2 Black“Scholes implied volatility 73

8.1.3 Black“Scholes implied volatility smile 74

8.1.4 Explanations for the ˜smile™ 75

8.2 Black“Scholes and no-arbitrage pricing 77

8.2.1 The stock price dynamics 77

8.2.2 The Black“Scholes partial differential equation 77

8.2.3 Solving the partial differential equation 79

8.3 Binomial method 80

8.3.1 Matching volatility with u and d 83

8.3.2 A two-step binomial tree and American-style

options 85

8.4 Testing option pricing model in practice 86

8.5 Dividend and early exercise premium 88

8.5.1 Known and ¬nite dividends 88

8.5.2 Dividend yield method 88

8.5.3 Barone-Adesi and Whaley quadratic

approximation 89

8.6 Measurement errors and bias 90

8.6.1 Investor risk preference 91

8.7 Appendix: Implementing Barone-Adesi and Whaley™s

ef¬cient algorithm 92

9 Option Pricing with Stochastic Volatility 97

9.1 The Heston stochastic volatility option pricing model 98

9.2 Heston price and Black“Scholes implied 99

9.3 Model assessment 102

9.3.1 Zero correlation 103

9.3.2 Nonzero correlation 103

9.4 Volatility forecast using the Heston model 105

9.5 Appendix: The market price of volatility risk 107

9.5.1 Ito™s lemma for two stochastic variables 107

9.5.2 The case of stochastic volatility 107

9.5.3 Constructing the risk-free strategy 108

x Contents

9.5.4 Correlated processes 110

9.5.5 The market price of risk 111

10 Option Forecasting Power 115

10.1 Using option implied standard deviation to forecast

volatility 115

10.2 At-the-money or weighted implied? 116

10.3 Implied biasedness 117

10.4 Volatility risk premium 119

11 Volatility Forecasting Records 121

11.1 Which volatility forecasting model? 121

11.2 Getting the right conditional variance and forecast

with the ˜wrong™ models 123

11.3 Predictability across different assets 124

11.3.1 Individual stocks 124

11.3.2 Stock market index 125

11.3.3 Exchange rate 126

11.3.4 Other assets 127

12 Volatility Models in Risk Management 129

12.1 Basel Committee and Basel Accords I & II 129

12.2 VaR and backtest 131

12.2.1 VaR 131

12.2.2 Backtest 132

12.2.3 The three-zone approach to backtest

evaluation 133

12.3 Extreme value theory and VaR estimation 135

12.3.1 The model 136

12.3.2 10-day VaR 137

12.3.3 Multivariate analysis 138

12.4 Evaluation of VaR models 139

13 VIX and Recent Changes in VIX 143

13.1 New de¬nition for VIX 143

13.2 What is the VXO? 144

13.3 Reason for the change 146

14 Where Next? 147

Contents xi

Appendix 149

References 201

Index 215

Foreword

If one invests in a ¬nancial asset today the return received at some pre-

speci¬ed point in the future should be considered as a random variable.

Such a variable can only be fully characterized by a distribution func-

tion or, more easily, by a density function. The main, single and most

important feature of the density is the expected or mean value, repre-

senting the location of the density. Around the mean is the uncertainty or

the volatility. If the realized returns are plotted against time, the jagged

oscillating appearance illustrates the volatility. This movement contains

both welcome elements, when surprisingly large returns occur, and also

certainly unwelcome ones, the returns far below the mean. The well-

known fact that a poor return can arise from an investment illustrates

the fact that investing can be risky and is why volatility is sometimes

equated with risk.

Volatility is itself a stock variable, having to be measured over a period

of time, rather than a ¬‚ow variable, measurable at any instant of time.

Similarly, a stock price is a ¬‚ow variable but a return is a stock variable.

Observed volatility has to be observed over stated periods of time, such

as hourly, daily, or weekly, say.

Having observed a time series of volatilities it is obviously interesting

to ask about the properties of the series: is it forecastable from its own

past, do other series improve these forecasts, can the series be mod-

eled conveniently and are there useful multivariate generalizations of

the results? Financial econometricians have been very inventive and in-

dustrious considering such questions and there is now a substantial and

often sophisticated literature in this area.

The present book by Professor Ser-Huang Poon surveys this literature

carefully and provides a very useful summary of the results available.

xiv Foreword

By so doing, she allows any interested worker to quickly catch up with

the ¬eld and also to discover the areas that are still available for further

exploration.

Clive W.J. Granger

December 2004

Preface

Volatility forecasting is crucial for option pricing, risk management and

portfolio management. Nowadays, volatility has become the subject of

trading. There are now exchange-traded contracts written on volatility.

Financial market volatility also has a wider impact on ¬nancial regula-

tion, monetary policy and macroeconomy. This book is about ¬nancial

market volatility forecasting. The aim is to put in one place models, tools

and ¬ndings from a large volume of published and working papers from

many experts. The material presented in this book is extended from two

review papers (˜Forecasting Financial Market Volatility: A Review™ in

the Journal of Economic Literature, 2003, 41, 2, pp. 478“539, and ˜Prac-

tical Issues in Forecasting Volatility™ in the Financial Analysts Journal,

2005, 61, 1, pp. 45“56) jointly published with Clive Granger.

Since the main focus of this book is on volatility forecasting perfor-

mance, only volatility models that have been tested for their forecasting

performance are selected for further analysis and discussion. Hence, this

book is oriented towards practical implementations. Volatility models

are not pure theoretical constructs. The practical importance of volatil-

ity modelling and forecasting in many ¬nance applications means that

the success or failure of volatility models will depend on the charac-

teristics of empirical data that they try to capture and predict. Given

the prominent role of option price as a source of volatility forecast, I

have also devoted much effort and the space of two chapters to cover

Black“Scholes and stochastic volatility option pricing models.

This book is intended for ¬rst- and second-year ¬nance PhD students

and practitioners who want to implement volatility forecasting models

but struggle to comprehend the huge volume of volatility research. Read-

ers who are interested in more technical aspects of volatility modelling

xvi Preface

could refer to, for example, Gourieroux (1997) on ARCH models,

Shephard (2003) on stochastic volatility and Fouque, Papanicolaou and

Sircar (2000) on stochastic volatility option pricing. Books that cover

speci¬c aspects or variants of volatility models include Franses and van

Dijk (2000) on nonlinear models, and Beran (1994) and Robinson (2003)

on long memory models. Specialist books that cover ¬nancial time se-

ries modelling in a more general context include Alexander (2001),

Tsay (2002) and Taylor (2005). There are also a number of edited series

that contain articles on volatility modelling and forecasting, e.g. Rossi

(1996), Knight and Satchell (2002) and Jarrow (1998).

I am very grateful to Clive for his teaching and guidance in the last

few years. Without his encouragement and support, our volatility survey

works and this book would not have got started. I would like to thank all

my co-authors on volatility research, in particular Bevan Blair, Namwon

Hyung, Eric Jondeau, Martin Martens, Michael Rockinger, Jon Tawn,

Stephen Taylor and Konstantinos Vonatsos. Much of the writing here

re¬‚ects experience gained from joint work with them.

1

Volatility De¬nition and

Estimation

1.1 WHAT IS VOLATILITY?

It is useful to start with an explanation of what volatility is, at least

for the purpose of clarifying the scope of this book. Volatility refers

to the spread of all likely outcomes of an uncertain variable. Typically,

in ¬nancial markets, we are often concerned with the spread of asset

returns. Statistically, volatility is often measured as the sample standard

deviation

T

1

σ= (rt ’ µ)2 , (1.1)

T ’1 t=1

where rt is the return on day t, and µ is the average return over the T -day

period.

Sometimes, variance, σ 2 , is used also as a volatility measure. Since

variance is simply the square of standard deviation, it makes no differ-

ence whichever measure we use when we compare the volatility of two

assets. However, variance is much less stable and less desirable than

standard deviation as an object for computer estimation and volatility

forecast evaluation. Moreover standard deviation has the same unit of

measure as the mean, i.e. if the mean is in dollar, then standard devi-

ation is also expressed in dollar whereas variance will be expressed in

dollar square. For this reason, standard deviation is more convenient and

intuitive when we think about volatility.

Volatility is related to, but not exactly the same as, risk. Risk is associ-

ated with undesirable outcome, whereas volatility as a measure strictly

for uncertainty could be due to a positive outcome. This important dif-

ference is often overlooked. Take the Sharpe ratio for example. The

Sharpe ratio is used for measuring the performance of an investment by

comparing the mean return in relation to its ˜risk™ proxy by its volatility.

2 Forecasting Financial Market Volatility

The Sharpe ratio is de¬ned as

Average Risk-free interest

’

return, µ rate, e.g. T-bill rate

Sharpe ratio = .

Standard deviation of returns, σ

The notion is that a larger Sharpe ratio is preferred to a smaller one. An

unusually large positive return, which is a desirable outcome, could lead

to a reduction in the Sharpe ratio because it will have a greater impact

on the standard deviation, σ , in the denominator than the average return,

µ, in the numerator.

More importantly, the reason that volatility is not a good or perfect

measure for risk is because volatility (or standard deviation) is only

a measure for the spread of a distribution and has no information on

its shape. The only exception is the case of a normal distribution or a

lognormal distribution where the mean, µ, and the standard deviation,

σ , are suf¬cient statistics for the entire distribution, i.e. with µ and σ

alone, one is able to reproduce the empirical distribution.

This book is about volatility only. Although volatility is not the sole

determinant of asset return distribution, it is a key input to many im-

portant ¬nance applications such as investment, portfolio construction,

option pricing, hedging, and risk management. When Clive Granger and

I completed our survey paper on volatility forecasting research, there

were 93 studies on our list plus several hundred non-forecasting papers

written on volatility modelling. At the time of writing this book, the

number of volatility studies is still rising and there are now about 120

volatility forecasting papers on the list. Financial market volatility is a

˜live™ subject and has many facets driven by political events, macroecon-

omy and investors™ behaviour. This book will elaborate some of these

complexities that kept the whole industry of volatility modelling and

forecasting going in the last three decades. A new trend now emerging

is on the trading and hedging of volatility. The Chicago Board of Ex-

change (CBOE) for example has started futures trading on a volatility

index. Options on such futures contracts are likely to follow. Volatility

swap contracts have been traded on the over-the-counter market well

before the CBOE™s developments. Previously volatility was an input to

a model for pricing an asset or option written on the asset. It is now the

principal subject of the model and valuation. One can only predict that

volatility research will intensify for at least the next decade.

Volatility De¬nition and Estimation 3

1.2 FINANCIAL MARKET STYLIZED FACTS

To give a brief appreciation of the amount of variation across different

¬nancial assets, Figure 1.1 plots the returns distributions of a normally

(a) Normal N(0,1) (b) Daily returns on S&P100

Jan 1965 “ Jul 2003

’5 ’4 ’3 ’2 ’1

’4 ’3 ’2 ’1 0 1 2 3 4 5

0 1 2 3 4 5

(c) £ vs. yen daily exchange rate returns (d) Daily returns on Legal & General share

Sep 1971 “ Jul 2003 Jan 1969 “ Jul 2003

’4 ’3 ’2 ’1 ’10 ’5

0 1 2 3 4 0 5 10

(e) Daily returns on UK Small Cap Index (f) Daily returns on silver

Jan 1986 “ Jul 2003 Aug 1971 “ Jul 2003

’4 ’3 ’2 ’1 ’10 ’5

0 1 2 3 4 0 5 10

Figure 1.1 Distribution of daily ¬nancial market returns. (Note: the dotted line is

the distribution of a normal random variable simulated using the mean and standard

deviation of the ¬nancial asset returns)

4 Forecasting Financial Market Volatility

distributed random variable, and the respective daily returns on the US

Standard and Poor market index (S&P100),1 the yen“sterling exchange

rate, the share of Legal & General (a major insurance company in the

UK), the UK Index for Small Capitalisation Stocks (i.e. small compa-

nies), and silver traded at the commodity exchange. The normal distri-

bution simulated using the mean and standard deviation of the ¬nancial

asset returns is drawn on the same graph to facilitate comparison.

From the small selection of ¬nancial asset returns presented in Fig-

ure 1.1, we notice several well-known features. Although the asset re-

turns have different degrees of variation, most of them have long ˜tails™ as

compared with the normally distributed random variable. Typically, the

asset distribution and the normal distribution cross at least three times,

leaving the ¬nancial asset returns with a longer left tail and a higher peak

in the middle. The implications are that, for a large part of the time, ¬nan-

cial asset returns ¬‚uctuate in a range smaller than a normal distribution.

But there are some occasions where ¬nancial asset returns swing in a

much wider scale than that permitted by a normal distribution. This phe-

nomenon is most acute in the case of UK Small Cap and silver. Table 1.1

provides some summary statistics for these ¬nancial time series.

The normally distributed variable has a skewness equal to zero and√

a kurtosis of 3. The annualized standard deviation is simply 252σ ,

assuming that there are 252 trading days in a year. The ¬nancial asset

returns are not adjusted for dividend. This omission is not likely to have

any impact on the summary statistics because the amount of dividends

distributed over the year is very small compared to the daily ¬‚uctuations

of asset prices. From Table 1.1, the Small Cap Index is the most nega-

tively skewed, meaning that it has a longer left tail (extreme losses) than

right tail (extreme gains). Kurtosis is a measure for tail thickness and

it is astronomical for S&P100, Small Cap Index and silver. However,

these skewness and kurtosis statistics are very sensitive to outliers. The

skewness statistic is much closer to zero, and the amount of kurtosis

dropped by 60% to 80%, when the October 1987 crash and a small

number of outliers are excluded.

Another characteristic of ¬nancial market volatility is the time-

varying nature of returns ¬‚uctuations, the discovery of which led to

Rob Engle™s Nobel Prize for his achievement in modelling it. Figure 1.2

plots the time series history of returns of the same set of assets presented

1

The data for S&P100 prior to 1986 comes from S&P500. Adjustments were made when the two series were

grafted together.

Table 1.1 Summary statistics for a selection of ¬nancial series

N (0, 1) S&P100 Yen/£ rate Legal & General UK Small Cap Silver

Start date Jan 65 Sep 71 Jan 69 Jan 86 Aug 71

Number of observations 8000 9675 7338 7684 4432 7771

Daily averagea 0 0.024 0.043 0.022 0.014

’0.021

Daily Standard Deviation 1 0.985 0.715 2.061 0.648 2.347

Annualized average 0 6.067 10.727 5.461 3.543

’5.188

Annualized Standard Deviation 15.875 15.632 11.356 32.715 10.286 37.255

Skewness 0 0.026 0.387

’1.337 ’0.523 ’3.099

Kurtosis 3 37.140 7.664 6.386 42.561 45.503

Number of outliers removed 1 5 9

Skewnessb ’0.055 ’0.917 ’0.088

Kurtosisb 7.989 13.972 15.369

aReturns not adjusted for dividends.

bThese two statistical measures are computed after the removal of outliers.

All series have an end date of 22 July, 2003.

(b) Daily returns on S&P100

10

(a) Normally distributed random variable N(0,1)

8

8

6

6

4

4 2

0

2

’2

0 ’4

’2 ’6

’8

’4

’10

’6

2

04 07 08 26 2 03 21 07 19 07

1

’8 01 02 02 01 01 12 12 11 11

5 9 3 70 1 5 88 92 96 00

96 96 97 97 98 98

1 1 1 1 1 1 19 19 19 20

’10

(c) Yen to £ exchange rate returns (d) Daily returns on Legal & General's share

6 20

4 15

2 10

0 5

0

’2

’4 ’5

’6 ’10

’8 ’15

1 3 1 5 5 1 7 2 6 5 4 1 1 8 1

06 14 17 06 11 02 15 15 02 24 10 18 05 16 04 15

83 90 22 61 71 92 10 22 31 50 61 53 51 41 40

0 01 06 10 04 10 03 06 10 02 04 06 07 09 09 10 10

10 40 70 10 30 51 71 00 20 40 60 80 00 20

7 7 7 79 8 69 71 73 76 78 81 83 85 88 90 92 94 96 98 00 02

98 98 98 99 99 99 99 99 00 00

19 19 19 19 19 1 1 1 1 1 1 1 1 2 2 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20

(f) Daily returns on silver

(e) Daily returns UK Small Cap Index 40

8

30

20

4

10

0

0

’10

’4

’20

’8

’30

’40

’12

2 0 6 1 7 4 1 0 8 3 0 0 8 7 4 0

8 4 6 0 0 0 3 3 5 5 4 7 7 0 7

10 31 51 72 92 20 21 42 62 90 11 12 32 60 81 81 91 91 93 03 11 21 22 10 10 11 11 11 21 31 40

60 70 80 90 00 11 30 40 50 60 71 90 00 10 20 10 30 50 70 91 11 31 51 80 00 20 40 60 80 00 37

98 98 98 98 99 99 99 99 99 99 99 99 00 00 00 97 97 97 97 97 98 98 98 98 99 99 99 99 99 00

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

Figure 1.2 Time series of daily returns on a simulated random variable and a collection of ¬nancial assets

Volatility De¬nition and Estimation 7

in Figure 1.1. The amplitude of the returns ¬‚uctuations represents the

amount of variation with respect to a short instance in time. It is clear

from Figures 1.2(b) to (f) that ¬‚uctuations of ¬nancial asset returns are

˜lumpier™ in contrast to the even variations of the normally distributed

variable in Figure 1.2(a). In the ¬nance literature, this ˜lumpiness™ is

called volatility clustering. With volatility clustering, a turbulent trad-

ing day tends to be followed by another turbulent day, while a tranquil

period tends to be followed by another tranquil period. Rob Engle (1982)

is the ¬rst to use the ARCH (autoregressive conditional heteroscedastic-

ity) model to capture this type of volatility persistence; ˜autoregressive™

because high/low volatility tends to persist, ˜conditional™ means time-

varying or with respect to a point in time, and ˜heteroscedasticity™ is a

technical jargon for non-constant volatility.2

There are several salient features about ¬nancial market returns and

volatility that are now well documented. These include fat tails and

volatility clustering that we mentioned above. Other characteristics doc-

umented in the literature include:

(i) Asset returns, rt , are not autocorrelated except possibly at lag one

due to nonsynchronous or thin trading. The lack of autocorrelation

pattern in returns corresponds to the notion of weak form market

ef¬ciency in the sense that returns are not predictable.

(ii) The autocorrelation function of |rt | and rt2 decays slowly and

corr (|rt | , |rt’1 |) > corr rt2 , rt’1 . The decay rate of the auto-

2

correlation function is much slower than the exponential rate of

a stationary AR or ARMA model. The autocorrelations remain

positive for very long lags. This is known as the long memory

effect of volatility which will be discussed in greater detail in

Chapter 5. In the table below, we give a brief taste of the ¬nding:

ρ(|r |) ρ(r 2 ) ρ(ln|r |) ρ(|T r |)

S&P100 35.687 3.912 27.466 41.930

Yen/£ 4.111 1.108 0.966 5.718

L&G 25.898 14.767 29.907 28.711

Small Cap 25.381 3.712 35.152 38.631

Silver 45.504 8.275 88.706 60.545

2

It is worth noting that the ARCH effect appears in many time series other than ¬nancial time series. In fact

Engle™s (1982) seminal work is illustrated with the UK in¬‚ation rate.

8 Forecasting Financial Market Volatility

(iii) The numbers reported above are the sum of autocorrelations for the

¬rst 1000 lags. The last column, ρ(|T r |), is the autocorrelation of

absolute returns after the most extreme 1% tail observations were

truncated. Let r0.01 and r0.99 be the 98% con¬dence interval of the

empirical distribution,

T r = Min [r, r0.99 ] , or Max [r, r0.01 ] . (1.2)

The effect of such an outlier truncation is discussed in Huber (1981).

The results reported in the table show that suppressing the large

numbers markedly increases the long memory effect.

(iv) Autocorrelation of powers of an absolute return are highest at power

one: corr (|rt | , |rt’1 |) > corr rtd , rt’1 , d = 1. Granger and Ding

d

(1995) call this property the Taylor effect, following Taylor (1986).

We showed above that other means of suppressing large numbers

could make the memory last longer. The absolute returns |rt | and

squared returns rt2 are proxies of daily volatility. By analysing the

more accurate volatility estimator, we note that the strongest auto-

correlation pattern is observed among realized volatility. Figure 1.3

demonstrates this convincingly.

(v) Volatility asymmetry: it has been observed that volatility increases if

the previous day returns are negative. This is known as the leverage

effect (Black, 1976; Christie, 1982) because the fall in stock price

causes leverage and ¬nancial risk of the ¬rm to increase. The phe-

nomenon of volatility asymmetry is most marked during large falls.

The leverage effect has not been tested between contemporaneous

returns and volatility possibly due to the fact that it is the previ-

ous day residuals returns (and its sign dummy) that are included

in the conditional volatility speci¬cation in many models. With the

availability of realized volatility, we ¬nd a similar, albeit slightly

weaker, relationship in volatility and the sign of contemporaneous

returns.

(vi) The returns and volatility of different assets (e.g. different company

shares) and different markets (e.g. stock vs. bond markets in one

or more regions) tend to move together. More recent research ¬nds

correlation among volatility is stronger than that among returns and

both tend to increase during bear markets and ¬nancial crises.

The art of volatility modelling is to exploit the time series proper-

ties and stylized facts of ¬nancial market volatility. Some ¬nancial time

series have their unique characteristics. The Korean stock market, for

Volatility De¬nition and Estimation 9

0.9

(a) Autocorrelation of daily returns on S&P100

0.7

0.5

0.3

0.1

’0.1

’0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

0.9

(b) Autocorrelation of daily squared returns on S&P100

0.7

0.5

0.3

0.1

’0.1

’0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

0.9 (c) Autocorrelation of daily absolute returns on S&P100

0.7

0.5

0.3

0.1

’0.1

’0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

(d) Autocorrelation of daily realized volatility of S&P100

0.9

0.7

0.5

0.3

0.1

’0.1

’0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Figure 1.3 Aurocorrelation of daily returns and proxies of daily volatility of S&P100.

(Note: dotted lines represent two standard errors)

example, clearly went through a regime shift with a much higher volatil-

ity level after 1998. Many of the Asian markets have behaved differently

since the Asian crisis in 1997. The dif¬culty and sophistication of volatil-

ity modelling lie in the controlling of these special and unique features

of each individual ¬nancial time series.

10 Forecasting Financial Market Volatility

1.3 VOLATILITY ESTIMATION

Consider a time series of returns rt , t = 1, · · · , T , the standard de-

viation, σ , in (1.1) is the unconditional volatility over the T period.

Since volatility does not remain constant through time, the conditional

volatility, σt,„ is a more relevant information for asset pricing and risk

management at time t. Volatility estimation procedure varies a great deal

depending on how much information we have at each sub-interval t, and

the length of „ , the volatility reference period. Many ¬nancial time series

are available at the daily interval, while „ could vary from 1 to 10 days

(for risk management), months (for option pricing) and years (for in-

vestment analysis). Recently, intraday transaction data has become more

widely available providing a channel for more accurate volatility esti-

mation and forecast. This is the area where much research effort has

been concentrated in the last two years.

When monthly volatility is required and daily data is available,

volatility can simply be calculated using Equation (1.1). Many macro-

economic series are available only at the monthly interval, so the current

practice is to use absolute monthly value to proxy for macro volatility.

The same applies to ¬nancial time series when a daily volatility estimate

is required and only daily data is available. The use of absolute value

to proxy for volatility is the equivalent of forcing T = 1 and µ = 0 in

Equation (1.1). Figlewski (1997) noted that the statistical properties of

the sample mean make it a very inaccurate estimate of the true mean es-

pecially for small samples. Taking deviations around zero instead of the

sample mean as in Equation (1.1) typically increases volatility forecast

accuracy.

The use of daily return to proxy daily volatility will produce a very

noisy volatility estimator. Section 1.3.1 explains this in a greater detail.

Engle (1982) was the ¬rst to propose the use of an ARCH (autoregres-

sive conditional heteroscedasticity) model below to produce conditional

volatility for in¬‚ation rate rt ;

rt = µ + µt , µt ∼ N 0, ht .

µt = z t h t ,

h t = ω + ±1 µt’1 + ±2 µt’2 + · · · .

2 2

(1.3)

The ARCH model is estimated by maximizing the likelihood of {µt }.

This approach of estimating conditional volatility is less noisy than the

absolute return approach but it relies on the assumption that (1.3) is the

Volatility De¬nition and Estimation 11

true return-generating process, µt is Gaussian and the time series is long

enough for such an estimation.

While Equation (1.1) is an unbiased estimator for σ 2 , the square root

of σ 2 is a biased estimator for σ due to Jensen inequality.3 Ding, Granger

and Engle (1993) suggest measuring volatility directly from absolute re-

turns. Davidian and Carroll (1987) show absolute returns volatility spec-

i¬cation is more robust against asymmetry and nonnormality. There is

some empirical evidence that deviations or absolute returns based mod-

els produce better volatility forecasts than models that are based on

squared returns (Taylor, 1986; Ederington and Guan, 2000a; McKenzie,

1999). However, the majority of time series volatility models, especially

the ARCH class models, are squared returns models. There are methods

for estimating volatility that are designed to exploit or reduce the in¬‚u-

ence of extremes.4 Again these methods would require the assumption

of a Gaussian variable or a particular distribution function for returns.

1.3.1 Using squared return as a proxy for daily volatility

Volatility is a latent variable. Before high-frequency data became widely

available, many researchers have resorted to using daily squared returns,

calculated from market daily closing prices, to proxy daily volatility.

Lopez (2001) shows that µt2 is an unbiased but extremely imprecise

estimator of σt2 due to its asymmetric distribution. Let

Yt = µ + µt , µt = σt z t , (1.4)

and z t ∼ N (0, 1). Then

E µt2 = σt2 E z t2 = σt2

t’1 t’1

since z t2 ∼ χ(1) . However, since the median of a χ(1) distribution is 0.455,

2 2

µt2 is less than 1 σt2 more than 50% of the time. In fact

2

1232 13

Pr µt2 ∈ σ, σ = Pr z t2 ∈ , = 0.2588,

2t 2t 22

which means that µt2 is 50% greater or smaller than σt2 nearly 75% of

the time!

√ √

If rt ∼ N 0, σt2 , then E (|rt |) = σt 2/π. Hence, σ t = |rt |/ 2/π if rt has a conditional normal distri-

3

bution.

4

For example, the maximum likelihood method proposed by Ball and Torous (1984), the high“low method

proposed by Parkinson (1980) and Garman and Klass (1980).

12 Forecasting Financial Market Volatility

Under the null hypothesis that returns in (1.4) are generated by a

GARCH(1,1) process, Andersen and Bollerslev (1998) show that the

population R 2 for the regression

µt2 = ± + βσ 2 + …t

t

is equal to κ ’1 where κ is the kurtosis of the standardized residuals and κ

is ¬nite. For conditional Gaussian error, the R 2 from a correctly speci¬ed

GARCH(1,1) model cannot be greater than 1/3. For thick tail distribu-

tion, the upper bound for R 2 is lower than 1/3. Christodoulakis and

Satchell (1998) extend the results to include compound normals and the

Gram“Charlier class of distributions con¬rming that the mis-estimation

of forecast performance is likely to be worsened by nonnormality known

to be widespread in ¬nancial data.

Hence, the use of µt2 as a volatility proxy will lead to low R 2 and under-

mine the inference on forecast accuracy. Blair, Poon and Taylor (2001)

report an increase of R 2 by three to four folds for the 1-day-ahead fore-

cast when intraday 5-minutes squared returns instead of daily squared

returns are used to proxy the actual volatility. The R 2 of the regression

of |µt | on σtintra is 28.5%. Extra caution is needed when interpreting em-

pirical ¬ndings in studies that adopt such a noisy volatility estimator.

Figure 1.4 shows the time series of these two volatility estimates over

the 7-year period from January 1993 to December 1999. Although the

overall trends look similar, the two volatility estimates differ in many

details.

1.3.2 Using the high“low measure to proxy volatility

The high“low, also known as the range-based or extreme-value, method

of estimating volatility is very convenient because daily high, low, open-

ing and closing prices are reported by major newspapers, and the cal-

culation is easy to program using a hand-held calculator. The high“low

volatility estimator was studied by Parkinson (1980), Garman and Klass

(1980), Beckers (1993), Rogers and Satchell (1991), Wiggins (1992),

Rogers, Satchell and Yoon (1994) and Alizadeh, Brandt and Diebold

(2002). It is based on the assumption that return is normally distributed

with conditional volatility σt . Let Ht and L t denote, respectively, the

highest and the lowest prices on day t. Applying the Parkinson (1980)

H -L measure to a price process that follows a geometric Brownian

Volatility De¬nition and Estimation 13

(a) Conditional variance proxied by daily squared returns

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

04/01/1993 04/01/1994 04/01/1995 04/01/1996 04/01/1997 04/01/1998 04/01/1999

0.08

(b) Conditional variance derived as the sum of intraday squared returns

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

04/01/1993 04/01/1994 04/01/1995 04/01/1996 04/01/1997 04/01/1998 04/01/1999

Figure 1.4 S&P100 daily volatility for the period from January 1993 to December

1999

motion results in the following volatility estimator (Bollen and Inder,

2002):

(ln Ht ’ ln L t )2

σ2

= t

4 ln 2

The Garman and Klass (1980) estimator is an extension of Parkinson

(1980) where information about opening, pt’1 , and closing, pt , prices

are incorporated as follows:

2 2

Ht pt

σ2 = 0.5 ln ’ 0.39 ln .

t

Lt pt’1

We have already shown that ¬nancial market returns are not likely to

be normally distributed and have a long tail distribution. As the H -L

volatility estimator is very sensitive to outliers, it will be useful to ap-

ply the trimming procedures in Section 1.4. Provided that there are no

destabilizing large values, the H -L volatility estimator is very ef¬cient

14 Forecasting Financial Market Volatility

and, unlike the realized volatility estimator introduced in the next sec-

tion, it is least affected by market microstructure effect.

1.3.3 Realized volatility, quadratic variation and jumps

More recently and with the increased availability of tick data, the term

realized volatility is now used to refer to volatility estimates calculated

using intraday squared returns at short intervals such as 5 or 15 minutes.5

For a series that has zero mean and no jumps, the realized volatility con-

verges to the continuous time volatility. To understand this, we assume

for the ease of exposition that the instantaneous returns are generated by

the continuous time martingale,

d p t = σt d W t , (1.5)

where d Wt denotes a standard Wiener process. From (1.5) the con-

ditional variance for the one-period returns, rt+1 ≡ pt+1 ’ pt , is

t+1 2

σs ds which is known as the integrated volatility over the period t

t

to t + 1. Note that while asset price pt can be observed at time t, the

volatility σt is an unobservable latent variable that scales the stochastic

process d Wt continuously through time.

Let m be the sampling frequency such that there are m continuously

compounded returns in one unit of time and

rm,t ≡ pt ’ pt’ 1/m (1.6)

and realized volatility

RVt+1 = rm,t+ j /m .

2

j=1,···,m

If the discretely sampled returns are serially uncorrelated and the sample

path for σt is continuous, it follows from the theory of quadratic variation

(Karatzas and Shreve, 1988) that

t+1

σs2 ds ’ = 0.

2

p lim rm,t+ j /m

m’∞ t j=1,···,m

Hence time t volatility is theoretically observable from the sample path

of the return process so long as the sampling process is frequent enough.

5

See Fung and Hsieh (1991) and Andersen and Bollerslev (1998). In the foreign exchange markets, quotes

for major exchange rates are available round the clock. In the case of stock markets, close-to-open squared return

is used in the volatility aggregation process during market close.

Volatility De¬nition and Estimation 15

When there are jumps in price process, (1.5) becomes

d pt = σt d Wt + κt dqt ,

where dqt is a Poisson process with dqt = 1 corresponding to a jump at

time t, and zero otherwise, and κt is the jump size at time t when there

is a jump. In this case, the quadratic variation for the cumulative return

process is then given by

t+1

σs2 ds + κ 2 (s) , (1.7)

t t<s¤t+1

which is the sum of the integrated volatility and jumps.

In the absence of jumps, the second term on the right-hand side of (1.7)

disappears, and the quadratic variation is simply equal to the integrated

volatility. In the presence of jumps, the realized volatility continues to

converge to the quadratic variation in (1.7)

m

t+1

σs2 ds + κ (s) ’ = 0.

2 2

p lim rm,t+ j /m (1.8)

m’∞ t t<s¤t+1 j=1

Barndorff-Nielsen and Shephard (2003) studied the property of the stan-

dardized realized bipower variation measure

m’1

a b

[a,b]

=m rm,t+ ( j+1)/m , a, b ≥ 0.

[(a+b)/2’1]

BVm,t+1 rm,t+ j /m

j=1

They showed that when jumps are large but rare, the simplest case where

a = b = 1,

m’1 t+1

µ’2 BVm,t+1 µ’2

[1,1]

= rm,t+ j /m rm,t+ ( j+1)/m ’ σs2 ds

1 1

t

j=1

√

where µ1 = 2/π . Hence, the realized volatility and the realized

bipower variation can be substituted into (1.8) to estimate the jump

component, κt . Barndorff-Nielsen and Shephard (2003) suggested im-

posing a nonnegative constraint on κt . This is perhaps too restrictive.

For nonnegative volatility, κt + µ’2 BVt > 0 will be suf¬cient.

1

Characteristics of ¬nancial market data suggest that returns measured

at an interval shorter than 5 minutes are plagued by spurious serial

correlation caused by various market microstructure effects including

nonsynchronous trading, discrete price observations, intraday periodic

16 Forecasting Financial Market Volatility

volatility patterns and bid“ask bounce.6 Bollen and Inder (2002), Ait-

Sahalia, Mykland and Zhang (2003) and Bandi and Russell (2004) have

given suggestions on how to isolate microstructure noise from realized

volatility estimator.

1.3.4 Scaling and actual volatility

The forecast of multi-period volatility σT,T + j (i.e. for j period) is taken to

be the sum of individual multi-step point forecasts s h T + j|T . These

j=1

multi-step point forecasts are produced by recursive substitution and

using the fact that µT +i|T = h T +i|T for i > 0 and µT +i|T = µT +i for

2 2 2

T + i ¤ 0. Since volatility of ¬nancial time series has complex struc-

ture, Diebold, Hickman, Inoue and Schuermann (1998) warn that fore-

cast estimates will differ depending on the current level of volatility,

volatility structure (e.g. the degree of persistence and mean reversion

etc.) and the forecast horizon.

If returns are iid (independent and identically distributed, or strict

white noise), then variance of returns over a long horizon can be derived

as a simple multiple of single-period variance. But, this is clearly not the

case for many ¬nancial time series because of the stylized facts listed in

Section 1.2. While a point forecast of σ T ’1,T | t’1 becomes very noisy

as T ’ ∞, a cumulative forecast, σ t,T | t’1 , becomes more accurate

because of errors cancellation and volatility mean reversion except when

there is a fundamental change in the volatility level or structure.7

Complication in relation to the choice of forecast horizon is partly

due to volatility mean reversion. In general, volatility forecast accu-

racy improves as data sampling frequency increases relative to forecast

horizon (Andersen, Bollerslev and Lange, 1999). However, for forecast-

ing volatility over a long horizon, Figlewski (1997) ¬nds forecast error

doubled in size when daily data, instead of monthly data, is used to fore-

cast volatility over 24 months. In some cases, where application is of

very long horizon e.g. over 10 years, volatility estimate calculated using

6

The bid“ask bounce for example induces negative autocorrelation in tick data and causes the realized

volatility estimator to be upwardly biased. Theoretical modelling of this issue so far assumes the price process

and the microstructure effect are not correlated, which is open to debate since market microstructure theory

suggests that trading has an impact on the ef¬cient price. I am grateful to Frank de Jong for explaining this to me

at a conference.

σ t,T | t’1 denotes a volatility forecast formulated at time t ’ 1 for volatility over the period from t to T . In

7

pricing options, the required volatility parameter is the expected volatility over the life of the option. The pricing

model relies on a riskless hedge to be followed through until the option reaches maturity. Therefore the required

volatility input, or the implied volatility derived, is a cumulative volatility forecast over the option maturity and

not a point forecast of volatility at option maturity. The interest in forecasting σ t,T | t’1 goes beyond the riskless

hedge argument, however.

Volatility De¬nition and Estimation 17

weekly or monthly data is better because volatility mean reversion is

dif¬cult to adjust using high frequency data. In general, model-based

forecasts lose supremacy when the forecast horizon increases with re-

spect to the data frequency. For forecast horizons that are longer than

6 months, a simple historical method using low-frequency data over a

period at least as long as the forecast horizon works best (Alford and

Boatsman, 1995; and Figlewski, 1997).

As far as sampling frequency is concerned, Drost and Nijman (1993)

prove, theoretically and for a special case (i.e. the GARCH(1,1) process,

which will be introduced in Chapter 4), that volatility structure should be

preserved through intertemporal aggregation. This means that whether

one models volatility at hourly, daily or monthly intervals, the volatility

structure should be the same. But, it is well known that this is not the

case in practice; volatility persistence, which is highly signi¬cant in

daily data, weakens as the frequency of data decreases. 8 This further

complicates any attempt to generalize volatility patterns and forecasting

results.

1.4 THE TREATMENT OF LARGE NUMBERS

In this section, I use large numbers to refer generally to extreme values,

outliers and rare jumps, a group of data that have similar characteristics

but do not necessarily belong to the same set. To a statistician, there are

always two ˜extremes™ in each sample, namely the minimum and the

maximum. The H -L method for estimating volatility described in the

previous section, for example, is also called the extreme value method.

We have also noted that these H -L estimators assume conditional dis-

tribution is normal. In extreme value statistics, normal distribution is but

one of the distributions for the tail. There are many other extreme value

distributions that have tails thinner or thicker than the normal distribu-

tion™s. We have known for a long time now that ¬nancial asset returns are

not normally distributed. We also know the standardized residuals from

ARCH models still display large kurtosis (see McCurdy and Morgan,

1987; Milhoj, 1987; Hsieh, 1989; Baillie and Bollerslev, 1989). Con-

ditional heteroscedasticity alone could not account for all the tail thick-

ness. This is true even when the Student-t distribution is used to construct

8

See Diebold (1988), Baillie and Bollerslev (1989) and Poon and Taylor (1992) for examples. Note that

Nelson (1992) points out separately that as the sampling frequency becomes shorter, volatility modelled using

discrete time model approaches its diffusion limit and persistence is to be expected provided that the underlying

returns is a diffusion or a near-diffusion process with no jumps.

18 Forecasting Financial Market Volatility

the likelihood function (see Bollerslev, 1987; Hsieh, 1989). Hence, in the

literature, the extreme values and the tail observations often refer to those

data that lie outside the (conditional) Gaussian region. Given that jumps

are large and are modelled as a separate component to the Brownian

motion, jumps could potentially be seen as a set similar to those tail

observations provided that they are truly rare.

Outliers are by de¬nition unusually large in scale. They are so large

that some have argued that they are generated from a completely dif-

ferent process or distribution. The frequency of occurrence should be

much smaller for outliers than for jumps or extreme values. Outliers

are so huge and rare that it is very unlikely that any modelling effort

will be able to capture and predict them. They have, however, undue

in¬‚uence on modelling and estimation (Huber, 1981). Unless extreme

value techniques are used where scale and marginal distribution are of-

ten removed, it is advisable that outliers are removed or trimmed before

modelling volatility. One such outlier in stock market returns is the Oc-

tober 1987 crash that produced a 1-day loss of over 20% in stock markets

worldwide.

The ways that outliers have been tackled in the literature largely de-

pend on their sizes, the frequency of their occurrence and whether these

outliers have an additive or a multiplicative impact. For the rare and

additive outliers, the most common treatment is simply to remove them

from the sample or omit them in the likelihood calculation (Kearns and

Pagan, 1993). Franses and Ghijsels (1999) ¬nd forecasting performance

of the GARCH model is substantially improved in four out of ¬ve stock

markets studied when the additive outliers are removed. For the rare

multiplicative outliers that produced a residual impact on volatility, a

dummy variable could be included in the conditional volatility equation

after the outlier returns has been dummied out in the mean equation

(Blair, Poon and Taylor, 2001).

rt = µ + ψ1 Dt + µt , µt = h t z t

h t = ω + βh t’1 + ±µt’1 + ψ2 Dt’1

2

where Dt is 1 when t refers to 19 October 1987 and 0 otherwise. Per-

sonally, I ¬nd a simple method such as the trimming rule in (1.2) very

quick to implement and effective.

The removal of outliers does not remove volatility persistence. In fact,

the evidence in the previous section shows that trimming the data using

(1.2) actually increases the ˜long memory™ in volatility making it appear

Volatility De¬nition and Estimation 19

to be extremely persistent. Since autocorrelation is de¬ned as

Cov (rt , rt’„ )

ρ (rt , rt’„ ) = ,

V ar (rt )

the removal of outliers has a great impact on the denominator, reduces

V ar (rt ) and increases the individual and the cumulative autocorrelation

coef¬cients.

Once the impact of outliers is removed, there are different views about

how the extremes and jumps should be handled vis-` -vis the rest of the

a

data. There are two schools of thought, each proposing a seemingly

different model, and both can explain the long memory in volatility. The

¬rst believes structural breaks in volatility cause mean level of volatility

to shift up and down. There is no restriction on the frequency or the size

of the breaks. The second advocates the regime-switching model where

volatility switches between high and low volatility states. The means of

the two states are ¬xed, but there is no restriction on the timing of the

switch, the duration of each regime and the probability of switching.

Sometimes a three-regime switching is adopted but, as the number of

regimes increases, the estimation and modelling become more complex.

Technically speaking, if there are in¬nite numbers of regimes then there

is no difference between the two models. The regime-switching model

and the structural break model will be described in Chapter 5.

2

Volatility Forecast Evaluation

Comparing forecasting performance of competing models is one of the

most important aspects of any forecasting exercise. In contrast to the

efforts made in the construction of volatility models and forecasts, little

attention has been paid to forecast evaluation in the volatility forecasting

literature. Let X t be the predicted variable, X t be the actual outcome and

µt = X t ’ X t be the forecast error. In the context of volatility forecast,

X t and X t are the predicted and actual conditional volatility. There are

many issues to consider:

(i) The form of X t : should it be σt2 or σt ?

(ii) Given that volatility is a latent variable, the impact of the noise

introduced in the estimation of X t , the actual volatility.

(iii) Which form of µt is more relevant for volatility model selection;

µt2 , |µt | or |µt |/X t ? Do we penalize underforecast, X t < X t ,

more than overforecast, X t > X t ?

(iv) Given that all error statistics are subject to noise, how do we know

if one model is truly better than another?

(v) How do we take into account when X t and X t+1 (and similarly for

µt and X t ) cover a large amount of overlapping data and are serially

correlated?

All these issues will be considered in the following sections.

2.1 THE FORM OF Xt

Here we argue that X t should be σt , and that if σt cannot be estimated

with some accuracy it is best not to perform comparison across predictive

models at all. The practice of using daily squared returns to proxy daily

conditional variance has been shown time and again to produce wrong

signals in model selection.

Given that all time series volatility models formulate forecasts based

on past information, they are not designed to predict shocks that are new

22 Forecasting Financial Market Volatility

to the system. Financial market volatility has many stylized facts. Once

a shock has entered the system, the merit of the volatility model depends

on how well it captures these stylized facts in predicting the volatility of

the following days. Hence we argue that X t should be σt . Conditional

variance σt2 formulation gives too much weight to the errors caused by

˜new™ shocks and especially the large ones, distorting the less extreme

forecasts where the models are to be assessed.

Note also that the square of a variance error is the fourth power of

the same error measured from standard deviation. This can complicate

the task of forecast evaluation given the dif¬culty in estimating fourth

moments with common distributions let alone the thick-tailed ones in

¬nance. The con¬dence interval of the mean error statistic can be very

wide when forecast errors are measured from variances and worse if they

are squared. This leads to dif¬culty in ¬nding signi¬cant differences

between forecasting models.

Davidian and Carroll (1987) make similar observations in their study

of variance function estimation for heteroscedastic regression. Using

high-order theory, they show that the use of square returns for modelling

variance is appropriate only for approximately normally distributed data,

and becomes nonrobust when there is a small departure from normal-

ity. Estimation of the variance function that is based on logarithmic

transformation or absolute returns is more robust against asymmetry

and nonnormality.

Some have argued that perhaps X t should be lnσt to rescale the size of

the forecast errors (Pagan and Schwert, 1990). This is perhaps one step

too far. After all, the magnitude of the error directly impacts on option

pricing, risk management and investment decision. Taking the logarithm

of the volatility error is likely to distort the loss function which is directly

proportional to the magnitude of forecast error. A decision maker might

be more risk-averse towards the larger errors.

We have explained in Section 1.3.1 the impact of using squared returns

to proxy daily volatility. Hansen and Lunde (2004b) used a series of

simulations to show that ˜. . . the substitution of a squared return for

the conditional variance in the evaluation of ARCH-type models can

result in an inferior model being chosen as [the] best with a probability

converges to one as the sample size increases . . . ™. Hansen and Lunde

(2004a) advocate the use of realized volatility in forecast evaluation but

caution the noise introduced by market macrostructure when the intraday

returns are too short.

Volatility Forecast Evaluation 23

2.2 ERROR STATISTICS AND THE FORM OF µt

Ideally an evaluation exercise should re¬‚ect the relative or absolute use-

fulness of a volatility forecast to investors. However, to do that one

needs to know the decision process that require these forecasts and the

costs and bene¬ts that result from using better forecasts. Utility-based

criteria, such as that used in West, Edison and Cho (1993), require some

assumptions about the shape and property of the utility function. In prac-

tice these costs, bene¬ts and utility function are not known and one often

resorts to simply use measures suggested by statisticians.

Popular evaluation measures used in the literature include

Mean Error (ME)

N N

1 1

µt = (σ t ’ σt ) ,

N N

t=1 t=1

Mean Square Error (MSE)

N N

1 1

µt2 = (σ t ’ σt )2 ,

N N

t=1 t=1

Root Mean Square Error (RMSE)

N N

1 1

µt2 = (σ t ’ σt )2 ,

N N

t=1 t=1

Mean Absolute Error (MAE)

N N

1 1

|µt | = |σ t ’ σt | ,

N N

t=1 t=1

Mean Absolute Percent Error (MAPE)

|µt | |σ t ’ σt |

N N

1 1

= .

σt σt

N N

t=1 t=1

Bollerslev and Ghysels (1996) suggested a heteroscedasticity-

adjusted version of MSE called HMSE where

2

N

σt

1

HMSE = ’1

σt

N t=1

24 Forecasting Financial Market Volatility

This is similar to squared percentage error but with the forecast error

scaled by predicted volatility. This type of performance measure is not

appropriate if the absolute magnitude of the forecast error is a major

concern. It is not clear why it is the predicted and not the actual volatility

that is used in the denominator. The squaring of the error again will give

greater weight to large errors.

Other less commonly used measures include mean logarithm of ab-

solute errors (MLAE) (as in Pagan and Schwert, 1990), the Theil-U

statistic and one based on asymmetric loss function, namely LINEX:

Mean Logarithm of Absolute Errors (MLAE)

N N

1 1

ln |µt | = ln |σ t ’ σt |

N N

t=1 t=1

Theil-U measure

N

(σ t ’ σt )2

t=1

Theil-U = , (2.1)

N

2

σ tB M ’ σt

t=1

where σ tB M is the benchmark forecast, used here to remove the effect of

any scalar transformation applied to σt .

LINEX has asymmetric loss function whereby the positive errors are

weighted differently from the negative errors:

N

1

[exp {’a (σ t ’ σt )} + a (σ t ’ σt ) ’ 1].

LINEX = (2.2)

N t=1

The choice of the parameter a is subjective. If a > 0, the function is

approximately linear for overprediction and exponential for underpre-

diction. Granger (1999) describes a variety of other asymmetric loss

functions of which the LINEX is an example. Given that most investors

would treat gains and losses differently, the use of asymmetric loss func-

tions may be advisable, but their use is not common in the literature.

2.3 COMPARING FORECAST ERRORS

OF DIFFERENT MODELS

In the special case where the error distribution of one forecasting

model dominates that of another forecasting model, the comparison is

Volatility Forecast Evaluation 25

straightforward (Granger, 1999). In practice, this is rarely the case, and

most comparisons of forecasting results are made based on the error

statistics described in Section 2.2. It is important to note that these er-

ror statistics are themselves subject to error and noise. So if an error

statistic of model A is higher than that of model B, one cannot conclude

that model B is better than A without performing tests of signi¬cance.

For statistical inference, West (1996), West and Cho (1995) and West

and McCracken (1998) show how standard errors for ME, MSE, MAE

and RMSE may be derived taking into account serial correlation in the

forecast errors and uncertainty inherent in volatility model parameter

estimates.

If there are T number of observations in the sample and T is large,

there are two ways in which out-of-sample forecasts may be made.

Assume that we use n number of observations for estimation and make

T ’ n number of forecasts. The recursive scheme starts with the sample

{1, · · · , n} and makes ¬rst forecast at n + 1. The second forecast for

n + 2 will include the last observation and form the information set

{1, · · · , n + 1}. It follows that the last forecast for T will include all

but the last observation, i.e. the information set is {1, · · · , T ’ 1}. In

practice, the rolling scheme is more popular, where a ¬xed number of

observations is used in the estimation. So the forecast for n + 2 will be

based on information set {2, · · · , n + 1}, and the last forecast at T will be

based on {T ’ n, · · · , T ’ 1}. The rolling scheme omits information in

the distant past. It is also more manageable in terms of computation when

T is very large. The standard errors developed by West and co-authors

are based on asymptotic theory and work for recursive scheme only. For

smaller sample and rolling scheme forecasts, Diebold and Mariano™s

(1995) small sample methods are more appropriate.

Diebold and Mariano (1995) propose three tests for ˜equal accuracy™

between two forecasting models. The tests relate prediction error to

some very general loss function and analyse loss differential derived

from errors produced by two competing models. The three tests include

an asymptotic test that corrects for series correlation and two exact ¬-

nite sample tests based on sign test and the Wilcoxon sign-rank test.

Simulation results show that the three tests are robust against non-

Gaussian, nonzero mean, serially and contemporaneously correlated

forecast errors. The two sign-based tests in particular continue to work

well among small samples. The Diebold and Mariano tests have been

used in a number of volatility forecasting contests. We provide the test

details here.

26 Forecasting Financial Market Volatility

Let { X it }t=1 and { X jt }t=1 be two sets of forecasts for {X t }t=1 from

T

T T

models i and j respectively. Let the associated forecast errors be {eit }t=1

T

and {e jt }t=1 . Let g (·) be the loss function (e.g. the error statistics in

T

Section 2.2) such that

g X t , X it = g (eit ) .

Next de¬ne loss differential

dt ≡ g (eit ) ’ g e jt .

The null hypothesis is equal forecast accuracy and zero loss differential

E(dt ) = 0.

2.3.1 Diebold and Mariano™s asymptotic test

The ¬rst test targets on the mean

T

1

d= |g(eit ) ’ g(e jt )|

T t=1

with test statistic

d

S1 = S1 ∼ N (0, 1)

1

2π f d (0)

T

T ’1

„

2π f d (0) = γ d („ )

1

S (T )

„ =’(T ’1)

T

1

γ d („ ) = dt ’ d dt’|„ | ’ d .

T t=|„ |+1

The operator 1 („ /S (T )) is the lag window, and S (T ) is the truncation

lag with

±